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AN INTRODUCTION TO QUANTUM FIELD THEORY ON CURVEDSPACE-TIMES

C. GÉRARD

1. Introduction

The purpose of these notes is to give an introduction to some recent aspects of QuantumField Theory on curved space-times, emphasizing its relations with partial differentialequations and microlocal analysis.

1.1. Quantum Field Theory. Quantum Field Theory arose from the need to unifyQuantum Mechanics with special relativity. However trying to treat the two basic relati-vistic field equations, the Klein-Gordon equation :

∂2t φ(t, x)−∆xφ(t, x) +m2φ(t, x) = 0, (t, x) ∈ R1+d,

and the Dirac equation :

γ0∂tψ(t, x) + γi∂xiψ(t, x)−mψ(t, x) = 0 (t, x) ∈ R1+d,

(where the γi are the Dirac matrices) in a way parallel to the non-relativistic Schroedingerequation :

∂tψ(t, x)− i

2m∆xψ(t, x) + iV (x)ψ(t, x) = 0

leads to difficulties (see eg [BD]). For the Klein-Gordon equation, there exists a conservedscalar product :

〈φ1|φ2〉 = i

ˆRd∂tφ1(t, x)φ2(t, x)− φ1(t, x)φ2(t, x)dx

which is not positive definite, hence cannot lead to a probabilistic interpretation. Howeveron has

〈φ|i∂tφ〉 ≥ 0, (positivity of the energy).For the Dirac equation the situation is the opposite : the conserved scalar product

〈ψ1|ψ2〉 =

ˆRdψ1(t, x) · ψ2(t, x)dx

is positive, but〈ψ|i∂tψ〉 is indefinite.

Hence it is possible to give a quantum mechanical interpretation of Dirac’s equation, butthe Hamiltonian will be unbounded below. This last issue was solved by Dirac, first byintroducing the notion of the Dirac sea, then a few years later by interpreting negativeenergy states as wave functions for the recently discovered positron.

The reason behind these difficulties is that, although all these equations are partialdifferential equations, their nature is very different : the Klein-Gordon and Dirac equations

1

2 C. GÉRARD

are classical equations, while the Schroedinger equation is a quantum equation, obtainedby quantizing the classical Newton equation :

x(t) = −∇xV (x(t)), x ∈ Rn.

or equivalently the Hamilton equations :x(t) = ∂ξh(x(t), ξ(t)),

ξ(t) = −∂xh(x(t), ξ(t))

for the classical Hamiltonian :

h(x, ξ) =1

2ξ2 + V (x).

Let us denote by X = (x, ξ) the points in T ∗Rn and introduce the coordinate functions

q : X 7→ x, p : X 7→ ξ.

If Φ(t) : T ∗Rn → T ∗Rn is the flow of Hh and q(t) := q Φ(t), p(t) := p Φ(t) then :∂tq(t) = p(t),

∂tp(t) = −∇V (q(t)),

which is known as the Liouville equation. Note that

pj(t), qk(t) = δjk, pj(t), pk(t) = qj(t), qk(t) = 0,

where ·, · is the Poisson bracket. To quantize the Liouville equation means to find aHilbert space H and functions R 3 t 7→ p(t), q(t) with values in selfadjoint operators onH such that

(1.1)

[pj(t), iqk(t)] = δjk1l, [pj(t), ipk(t)] = [qj(t), iqk(t)] = 0,

∂tq(t) = p(t),

∂tp(t) = −∇V (q(t)).

The equations in the first time are called the (fixed time)canonical commutation relations.The last two equations are called Heisenberg equations. The solution is as follows :(1) Find operators p, q satisfying

[pj, iqk] = δjk1l, [pj, ipk] = [qj, iqk] = 0.

(2) Construct the following selfadjoint operator on H

H =1

2p2 + V (q).

Thenq(t) := eitHqe−itH , p(t) := eitHpe−itH

solve (1.1).The Stone-von Neumann theorem says that there is no choice in step (1) : modulo some

technical conditions and multiplicity one has only one choice, up to unitary equivalence :

H = L2(Rn), q = x, p = i−1∇x.

Then H = −12∆ +V (x) is the Schroedinger operator. step (2) is then a standard problem

in the theory of selfadjoint operators.

AN INTRODUCTION TO QUANTUM FIELD THEORY ON CURVED SPACE-TIMES 3

The Klein-Gordon equation is also a Hamiltonian equation, however with a infinitedimensional phase space, which can be taken for example as C∞0 (Rd) ⊕ C∞0 (Rd). Theclassical Hamiltonian is then

h(ϕ, π) :=1

2

ˆRdπ2(x) + |∇xϕ(x)|2 +m2ϕ2(x)dx,

for the linear case, or

h(ϕ, π) :=1

2

ˆRdπ2(x) + |∇xϕ(x)|2 +m2ϕ2(x) + ϕn(x)dx,

for some non-linear version. Here the symbols ϕ(x), π(x) are (coordinate) functions, pa-rametrized by a point x ∈ Rd, on the space of smooth solutions of the Klein-Gordonequation, with compactly supported Cauchy data. If φ is such a solution then

ϕ(x)(φ) := φ(0, x), π(x)(φ) := ∂tφ(0, x)

It is well-known that these are symplectic coordinates, i.e.

ϕ(x), ϕ(x′) = π(x), π(x′) = 0, π(x), φ(x′) = δ(x, x′), ∀ x, x′ ∈ Rd.

One would like to follow the same path and consider families of operators on a Hilbertspace H, ϕ(x), π(x), x ∈ Rd such that

[π(x), iϕ(x′)] = δ(x− x′)1l, [ϕ(x), iϕ(x′)] = [π(x), iπ(x′)] = 0, ∀ x, x′ ∈ Rd.

The fundamental difference with non-relativistic Quantum Mechanics is that, since thephase space is infinite dimensional, the Stone von Neumann theorem cannot be appliedanymore : there exists an infinite number of inequivalent representations of commutationrelations.

In other words, when one tries to quantize a classical field equation, the Hilbert spacehas to be constructed together with the quantum Hamiltonian : one cannot work on ourfamiliar Hilbert space and then use tools from operator theory to construct the quantumHamiltonian.

This is the reason why the rigorous construction of Quantum Field Theory models is sodifficult, except for non-interacting theories. For interacting theories it has been achievedonly in 2 and 3 spacetime dimensions, see the construction of the P (ϕ)2 and ϕ4

3 models,which were the landmark successes of constructive field theory. In 4 spacetime dimensionsone has to rely instead on perturbative methods.

Another lesson learned from Quantum Field Theory (and also from Quantum StatisticalMechanics), is that Hilbert spaces do not play such a central role anymore. Instead onefocuses on algebras and states.

Let us finish this discussion by recalling a well-known anecdote : at the Solvay conferencein 1927, Dirac told Bohr that he was trying to find a relativistic quantum theory ofthe electron (i.e. the Dirac equation). Bohr replied that this problem had already beensolved by Klein, who had found the Klein-Gordon equation. We know now that these twoequations are of a different nature, the first describing fermionic fields, the second bosonicones, and that they can be interpreted as quantum equations only via Quantum FieldTheory.

4 C. GÉRARD

1.2. QFT on curved space-times. Given the difficulties with the construction of inter-acting field theories on Minkowski space-time, one may wonder why one should considerquantum field theories on curved space-times, which have no reason to be simpler.

One reason comes from attempts to quantize gravitation, where one starts by linearizingEinstein equations around a curved background metric g. Another argument is that thereare several interesting quantum effects appearing in presence of strong gravitational fields.The most famous one is the Hawking effect, which predicts that a black hole can emitquantum particles.

There are several new challenges one has to face when moving from flat Minkowskispace-time to an arbitrary curved space-time.

On the computational side, one cannot rely anymore on the Fourier transform andrelated analyticity arguments, which are natural and useful on Minkowski space, sincethe Klein-Gordon equation has then constant coefficients.

On a more conceptual side, a curved space-time does not have the large group ofisometries (the Poincaré group) of the Minkowski space. It follows that on a curved space-time there seems to be no natural notion of a vacuum state, which is defined on Minkowskispace as the unique state which is invariant under space-time translations, and has anadditional positive energy condition.

In the eigthies, physicists managed to define a class of states, the so-called Hadamardstates, which were characterized by properties of their two-point functions, which had tohave a specific asymptotic expansion near the diagonal, connected with the Hadamardparametrix construction for the Klein-Gordon equation on a curved space-time.

Later in 1995, in a seminal paper, Radzikowski reformulated the old Hadamard condi-tion in terms of the wave front set of the two-point function. The wave front set of adistribution, introduced in 1970 by Hörmander, is one of the important notions of micro-local analysis, a theory which was precisely developed to extend Fourier analysis, in thestudy of general partial differential equations.

The introduction of tools from microlocal analysis had a great influence on the field,leading for example to the proof of renormalizability of scalar interacting field theories byBrunetti and Fredenhagen [BF].

The goal of these notes is to give an introduction to the modern notion of Hadamardstates, for a mathematically oriented audience.

2. A quick introduction to Quantum Mechanics

This section is supposed to give a very quick introduction to the mathematical formalismof Quantum Mechanics, which is (or is expected to be) still relevant to Quantum FieldTheory.

2.1. Hilbert space approach. In ordinary Quantum Mechanics, the description of aphysical system starts with a Hilbert space H, whose scalar product is denoted by (u|v).The states of the system are described by unit vectors ψ ∈ H with ‖ψ‖ = 1.

The various physical quantities which can be measured (like position, momentum,energy, spin) are represented by selfadjoint operators on H, i.e. (forgetting about impor-tant issues with unbounded operators), linear operators A on H, assumed to be boundedfor simplicity, such that A = A∗, (where A∗ is the adjoint of A), called observables.


If ψ ∈ H , ‖ψ‖ = 1 is a state vector, then the map :

A 7→ ωψ(A) = (ψ|Aψ)

computes the expectation value of A in the state ψ represent the average value of actualmeasurements of the physical quantity represented by A.

Rather quickly people were led to consider also mixed states, where the state of thesystem is only incompletely known. For example if ψi, i ∈ N is an orthonormal familyand 0 ≤ ρi ≤ 1 are real numbers with

∑∞i=0 ρi = 1, then we can consider the trace-class

operator :

ρ =∞∑i=0

ρi|ψi)(ψi|, Trρ = 1,

called a density matrix and the map

A 7→ ωρ(A) := Tr(ρA)

is called a mixed state. Vector states are also called pure states.

2.2. Algebraic approach. The framework above is sufficient to cover all of non-relativisticQuantum Mechanics, i.e. in practice quantum systems consisting of a finite number ofnon-relativistic particles. However when one considers systems with an infinite numberof particles, like in statistical mechanics, or quantum field theory, where the notion ofparticles is dubious, an algebraic framework is more relevant. It starts with the followingobservation about the space B(H) of bounded operators on H :

if we equip it with the operator norm, it is a Banach space, and a Banach algebra,i.e. an algebra with the property that ‖AB‖ ≤ ‖A‖‖B‖. It is also an involutive Banachalgebra, i.e. the adjoint operation A 7→ A∗ has the properties that

(AB)∗ = B∗A∗, ‖A∗‖ = ‖A‖.Finally one can easily check that :

‖A∗A‖ = ‖A‖2, A ∈ B(H).

This last property has very important consequences, for example one can deduce from itthe functional calculus and spectral theorem for selfadjoint operators.

An abstract algebra A equipped with a norm and an involution with these properties,which is moreover complete is called a C∗algebra. The typical example of a C∗ algebra isof course the algebra B(H) of bounded operators on a Hilbert space.

If H is a Hilbert space, a ∗−homomorphism

A 3 A 7→ π(A) ∈ B(H)

is callled a representation of A in H. An injective representation is called faithful.The need for such change of point of view comes from the fact that a physical system,

like a gas of electrons, can exist in many different physical realizations, for example atdifferent temperatures. In other words it does not come equipped with a canonical Hilbertspace.

Observables, like for example the electron density, have a meaning irrelevant of therealizations, and are described by selfadjoint elements in some C∗ algebra A. However

6 C. GÉRARD

the various Hilbert spaces and the representations of the observables on them are verydifferent from one temperature to another.

One can also describe the possible physical realizations of a system with the languageof states. A state ω on A is a linear map :

ω : A 7→ C

such thatω(A∗A) ≥ 0, A ∈ A.

Assuming that A has a unit (which can always be assumed by adjoining one), one alsorequires that

ω(1l) = 1.

The set of states on a C∗ algebra is a convex set, its extremal points are called pure states.If A ⊂ B(H) and ψ is a unit vector, or if ρ is a density matrix, then

ωψ(A) := (ψ|Aψ), ωρ(A) := Tr(ρA)

are states on A. If A = B(H), then ωψ is a pure state. It is important to be aware of thefact that if A is only a C∗ subalgebra of B(H), then ωψ may not be a pure state on A.

2.3. The GNS construction. After being told that one should use C∗ algebras andstates, one can wonder where the Hilbert spaces have gone. Actually given a C∗ algebraA and a state ω on it, it is quite easy to construct a canonical Hilbert space and arepresentation of A on it, as proved by Gelfand, Naimark and Segal. There exist a triple(Hω, πω,Ωω), unique up to unitary equivalence, where Hω is a Hilbert space, πω : A 7→B(Hω) is a faithful representation, Ωω ∈ Hω is a unit vector such that :

ω(A) = (Ωω|πω(A)Ωω), A ∈ A,

and πω(A)Ωω : A ∈ A is dense in Hω.

3. Notation

In this section we collect some notation that will be used in these notes. If X is a realor complex vector space we denote by X# its dual. Bilinear forms on X are identified withelements of L(X ,X#), which leads to the notation x1·bx2 for b ∈ L(X ,X#), x1, x2 ∈ X . Thespace of symmetric (resp. anti-symmetric) bilinear forms on X is denoted by Ls(X ,X#)(resp. La(X ,X#)).

If σ ∈ Ls(X ,X#) is non-degenerate, we denote byO(X , σ) the linear (pseudo-)orthogonalgroup on X . Similarly if σ ∈ La(X ,X#) is non-degenerate, i.e. (X , σ) is a symplectic space,we denote by Sp(X , σ) the linear symplectic group on X .

If Y is a complex vector space, we denote by YR its realification, i.e. Y considered asa real vector space. We denote by Y a conjugate vector space to Y , i.e. a complex vectorspace Y with an anti-linear isomorphism Y 3 y 7→ y ∈ Y . The canonical conjugate vectorspace to Y is simply the real vector space YR equipped with the complex structure −i, ifi is the complex structure of Y . In this case the map y → y is chosen as the identity. Ifa ∈ L(Y1,Y2), we denote by a ∈ L(Y1,Y2) the linear map defined by :

(3.1) ay1 := ay1, y1 ∈ Y1.


We denote by Y∗ the anti-dual of Y , i.e. the space of anti-linear forms on Y . ClearlyY∗ can be identified with Y# ∼ Y#.

Sesquilinear forms on Y are identified with elements of L(Y ,Y∗), and we use the nota-tion (y1|by2) or y1 ·by2 for b ∈ L(Y ,Y∗), y1, y2 ∈ Y .

The space of hermitian (resp. anti-hermitian) sesquilinear forms on Y is denoted byLs(Y ,Y∗) (resp. La(Y ,Y∗)).

If q ∈ Lh(Y ,Y∗) is non-degenerate, i.e. (Y , q) is a pseudo-unitary space, we denote byU(Y , q) the linear pseudo-unitary group on Y .

If b is a bilinear form on the real vector space X , its canonical sesquilinear extensionto CX is by definition the sesquilinear form bC on CX given by

(w1|bCw2) := x1 ·bx2 + y1 ·by2 + ix1 ·by2 − iy1 ·bx2, wi = xi + iyi

for xi, yi ∈ X , i = 1, 2. This extension maps (anti-)symmetric forms on X onto (anti-)hermitian forms on CX .

Conversely if Y is a complex vector space and YR is its realification, i.e. Y consideredas a real vector space, then for b ∈ Lh/a(Y ,Y∗) the form Reb belongs to Ls/a(YR,Y#

R ).

4. CCR and CAR algebras

4.1. Introduction. It is useful to discuss the CCR and CAR without making referenceto a Fock space. There are some mathematical subtleties with CCR algebras, coming fromthe fact that the field operators are ’unbounded’. These subtleties can mostly be ignoredfor our purposes.

4.2. Algebras generated by symbols and relations. In physics many algebras aredefined by specifying a set of generators and the relations they satisfy. This is completelysufficient to do computations, but mathematicians may feel uncomfortable with such anapproach. However it is easy (and actually rather useless) to give a rigorous definition.

Assume that A is a set. We denote by cc(A,K) the vector space of functions A → Kwith finite support (usually K = C). If for A ∈ A, we denote the indicator function 1lAsimply by A, we see that any element of cc(A,K) can be written as

∑A∈B λAA, B ⊂ A

finite, λA ∈ K.Then cc(A,K) can be seen as the vector space of finite linear combinations of elements

of A. We setA(A, 1l) :=

al⊗ cc(A,K),

called the universal unital algebra over K with generators A. Usually one write A1 · · ·Aninstead of A1 ⊗ · · · ⊗ An for Ai ∈ A.

Let us denote by A another copy of A. We denote by a the element a ∈ A. We set then∗a := a, ∗a := a and extend ∗ to A(A tA, 1l) by setting

(b1b2 · · · bn)∗ = b∗n · · · b∗2b∗1, bi ∈ A tA, 1l = 1l∗.

The algebra A(A t A, 1l) equipped with the involution ∗ is called the universal unital∗−algebra over K with generators A.

Let now R ⊂ A(A, 1l) (the set of ’relations’). We denote by I(R) the ideal of A(A,K)generated by R. Then the quotient

A(A, 1l)/I(R)

8 C. GÉRARD

is called the unital algebra with generators A and relations R = 0, R ∈ R.Similarly if R ⊂ A(A∪A, 1l) is ∗-invariant, then A(AtA, 1l)/I(R) is called the unital∗-algebra with generators A tA and relations R = 0, R ∈ R.

4.3. Polynomial CCR algebra. We fix a (real) presymplectic space (X , σ), i.e. σ ∈La(X ,X#) is not supposed to be injective.

Definition 4.1. The polynomial CCR ∗-algebra over X , denoted by CCRpol(X , σ), isdefined to be the unital complex ∗-algebra generated by elements φ(x), x ∈ X , with relations

φ(λx) = λφ(x), λ ∈ R, φ(x1 + x2) = φ(x1) + φ(x2),

(())φ∗(x) = φ(x), φ(x1)φ(x2)− φ(x2)φ(x1) = ix1·σx21l.

4.4. Weyl CCR algebra. One problem with CCRpol(X , σ) is that (unless σ = 0) itselements cannot be faithfully represented as bounded operators on a Hilbert space. Tocure this problem one has to work with Weyl operators, which lead to the Weyl CCR∗−algebra.

Definition 4.2. The algebraic Weyl CCR algebra over X denoted by CCRWeyl(X , σ) isthe ∗-algebra generated by the elements W (x), x ∈ X , with relations

W (0) = 1l, W (x)∗ = W (−x),

W (x1)W (x2) = e−i2x1·σx2W (x1 + x2), x, x1, x2 ∈ X .

It is possible to equip CCRWeyl(X , σ) with a unique C∗−norm. Its completion for thisnorm is called the Weyl CCR algebra over X and still denoted by CCRWeyl(X , σ). We willmostly work with CCRpol(X , σ), which will simply be denoted by CCR(X , σ). Of coursethe formal relation between the two approaches is

W (x) = eiφ(x), x ∈ X ,which does not make sense a priori, but from which mathematically correct statementscan be deduced.

4.5. Charged symplectic spaces.

Definition 4.3. A complex vector space Y equipped with a a non-degenerate anti-hermitiansesquilinear form σ is called a charged symplectic space. We set

q := iσ ∈ Lh(Y ,Y∗),which is called the charge.

4.6. Kähler spaces. Let (Y , σ) be a charged symplectic space. Its complex structure willbe denoted by j ∈ L(YR) (to distinguish it from the complex number i ∈ C). Note that(YR,Reσ) is a real symplectic space with j ∈ Sp(YR,Reσ) and j2 = −1l. We have :

y1qy2 = y1 · Reσjy2 + iy1 · Reσy2, y1, y2 ∈ Y .The converse construction is as follows : A real (pre-)symplectic space (X , σ) with a mapj ∈ L(X ) such that

j2 = −1l, j ∈ Sp(X , σ),


is called a pseudo-Kähler space. If in addition ν := σj is positive definite, it is called aKähler space. We set now

Y = (X , j),which is a complex vector space, whose elements are logically denoted by y. If (X , σ, j) isa pseudo-Kähler space we can set :

y1qy2 := y1 · σjy2 + iy1 · σy2, y1, y2 ∈ Y ,and check that q is sesquilinear hermitian on Y equipped with the complex structure j.One can consider the CCR algebra CCRpol(YR,Reσ), with selfadjoint generators φ(y) andrelations :

[φ(y1), φ(y2)] = iy1 · Reσy21l.

One can instead generate CCRpol(YR,Reσ) by the charged fields :

ψ(y) :=1√2

(φ(y) + iφ(jy)), ψ∗(y) :=1√2

(φ(y)− iφ(jy)), y ∈ Y .

The map Y 3 y 7→ ψ∗(y) (resp. Y 3 y 7→ ψ(y)) is C−linear (resp. C−anti-linear). Thecommutation relations take the form :

[ψ(y1), ψ(y2)] = [ψ∗(y1), ψ∗(y2)] = 0,

[ψ(y1), ψ∗(y2)] = y1 · qy21l, y1, y2 ∈ Y .Note the similarity with the CCR expressed in terms of creation/annihilation operators,the difference being the fact that q is not necessarily positive. In this context, it is naturalto denote CCR(YR,Reσ) by CCR(Y , σ).

4.7. CAR algebra. We fix an Euclidean space (X , ν) (possibly infinite dimensional).

Definition 4.4. The algebraic CAR algebra over X , denoted CARalg(X , ν), is the complexunital ∗-algebra generated by elements φ(x), x ∈ X , with relations

φ(λx) = λφ(x), λ ∈ R, φ(x1 + x2) = φ(x1) + φ(x2),

φ∗(x) = φ(x), φ(x1)φ(x2) + φ(x2)φ(x1) = 2x1·νx21l.

Again CARalg(X , ν) has a unique C∗−norm, and its completion is denoted by CAR(X , ν).

4.8. Kähler spaces. Let now (Y , ν) be a hermitian space, denoting again its complexstructure by j. Then (YR,Reν) is an euclidean space, with j ∈ U(YR,Reν). We have :

y1 · νy2 = y1 · Reνy2 − iy1 · Rejy2, y1, y2 ∈ Y .Denoting by φ(y) the selfadjoint fields which generate CAR(YR,Reν), we can introducethe charged fields :

ψ(y) := φ(y) + iφ(jy), ψ∗(y) := φ(y)− iφ(jy), y ∈ Y .Again the map Y 3 y 7→ ψ∗(y) (resp. Y 3 y 7→ ψ(y)) is C−linear (resp. C−anti-linear).The anti-commutation relations take the form :

[ψ(y1), ψ(y2)]+ = [ψ∗(y1), ψ∗(y2)]+ = 0,

[ψ(y1), ψ∗(y2)]+ = 2y1 · νy21l, y1, y2 ∈ Y .

10 C. GÉRARD

5. States on CCR/CAR algebras

5.1. Introduction. Let A be a unital ∗−algebra. A state on A is a linear map ω : A→ Ksuch that

ω(A∗A) ≥ 0, ∀A ∈ A, ω(1l) = 1.

Elements of the form A∗A are called positive.Let X be either a presymplectic or euclidean space and let ω be a state on CCR(X , σ)

or CAR(X , ν). One can associate to ω a bilinear form on X called the 2−point function :

X × X 3 (x1, x2) 7→ ω(φ(x1)φ(x2)).

Of course to completely specify the state ω one also needs to know the n−point functions :X n 3 (x1, . . . xn) 7→ ω(φ(x1) . . . φ(xn)).

A particularly useful class of states are the quasi-free states, which are defined by the factthat all n−point functions are determined by the 2−point function.

5.2. Bosonic quasi-free states. Let (X , σ) a presymplectic space and ω a state onCCRWeyl(X , σ). The function :

X 3 x 7→ ω(W (x)) =: G(x)

is called the characteristic function of the state ω, and is a non-commutative version ofthe Fourier transform of a probability measure.

There is also a non-commutative version of Bochner’s theorem (the theorem whichcharacterizes these Fourier transforms) :

Proposition 5.1. A map G : X → C is the characteristic function of a state onCCRWeyl(X , σ) iff for any n ∈ N, xi ∈ X the n× n matrix[

G(xj − xi)ei2xi·σxj

]1≤i,j≤n

is positive.

Proof. ⇒ : for x1, . . . , xn ∈ X , λ1, . . . , λn ∈ C set

A :=n∑j=1

λjW (xj) ∈ CCRWeyl(X , σ).

Such A are dense in CCRWeyl(X , σ). One computes A∗A using the CCR and obtains that :

A∗A =n∑

j,k=1

λjλkW (xj − xk)ei2xj ·σxk ,

from which ⇒ follows.⇐ : one uses exactly the same argument, defining ω using G, the above formula shows

that ω is positive. 2

Definition 5.2. (1) A state ω on CCRWeyl(X , σ) is a quasi-free state if there existsη ∈ Ls(X ,X#) (a symmetric form on X ) such that

(5.1) ω(W (x)

)= e−

12x·ηx, x ∈ X .


(2) The form η is called the covariance of the quasi-free state ω.

Quasi-free states should be considered as non-commutative versions of Gaussian mea-sures. To explain this remark, consider the Gaussian measure on Rd with covariance η

dµη := (2π)d/2 det η−1

2e−

12y·η−1ydy.

We have : ˆeix·ydµη(y) = e−

12x·ηx.

Note also that if xi ∈ Rd, then´ ∏2n+11 xi · ydµη(y) = 0,´ ∏2n1 xi · ydµη(y) =

∑σ∈Pair2n

∏nj=1 xσ(2j−1) · ηxσ(2j),

which should be compared with Def. 5.4 below.

Proposition 5.3. Let η ∈ Ls(X ,X#). Then the following are equivalent :(1) X 3 x 7→ e−

12x·ηx is a characteristic function and hence there exists a quasi-free state

satisfying (5.1).(2) ηC + i

2σC ≥ 0 on CX , where ηC, σC ∈ L

(CX , (CX )∗

)are the canonical sesquilinear

extensions of η, σ.(3) |x1·σx2| ≤ 2(x1·ηx1)

12 (x2·ηx2)

12 , x1, x2 ∈ X .

Proof. The proof of (1)⇒ (2) is easy, by considering complex fields φ(w) = φ(x1)+iφ(x2),w = x1 + ix2 and noting that the positivity of ω implies that ω(φ∗(w)φ(w)) ≥ 0, for anyw ∈ CX .

The proof of (2)⇒ (1) is more involved : let us fix x1, . . . , xn ∈ X and set

bjk = xj · ηxk +i

2xj · σxk.

Then, for λ1, . . . , λn ∈ C,∑1≤j,k≤n

λjbjkλk = w·ηCw + i2w·ωCw, w =

n∑j=1

λjxj ∈ CX .

By (2), the matrix [bjk] is positive. One has then to use an easy lemma, saying that thepointwise product of two positive matrices is positive. From this it follows also that [ebjk ]

is positive, and hence the matrix [e−12xj ·ηxjebjke−

12xk·ηxk ] is positive. Hence :∑n

j,k=1 e−12

(xk−xj)·η(xk−xj)ei2xj ·ωxkλjλk

=∑n

j,k=1 e−12xj ·ηxjebjke−

12xj ·ηxjλjλk

=∑n

j,k=1 G(xj − xk)ei2xj ·σxk ≥ 0,

for G(x) = e−12x·ηx. By Prop. 5.1, this means that G is a characteristic function. The proof

of (2)⇔ (3) is an exercise in linear algebra. 2

12 C. GÉRARD

It is easy to deduce from ω the corresponding state acting on CCRpol(X , σ), by setting :

ω(φ(x1) · · ·φ(xn))

:= ddt1· · · d

dtnω(W (t1x1 + · · ·+ tnxn))|t1=···tn=0.

In particular

ω(φ(x1)φ(x2)) = x1 · ηx2 +i

2x1 · σx2.

The corresponding definition of a quasi-free state on CCRpol(X , σ) is as follows :

Definition 5.4. A state ω on CCRpol(X , σ) is quasi-free if

ω(φ(x1) · · ·φ(x2m−1)

)= 0,

ω(φ(x1) · · ·φ(x2m)

)=

∑σ∈Pair2m

m∏j=1

ω(φ(xσ(2j−1))φ(xσ(2j)

).

We recall that Pair2m is the set of pairings, i.e. the set of partitions of 1, . . . , 2m intopairs. Any pairing can be written as

i1, j1, · · · , im, jmfor ik < jk and ik < ik+1, hence can be uniquely identified with a partition σ ∈ S2m suchthat σ(2k − 1) = ik, σ(2k) = jk.

5.3. Gauge-invariant quasi-free states. Let us now assume that (X , σ, j) is a pseudo-Kähler space, i.e. that there exists an anti-involution j ∈ Sp(X , σ). Note that we have

ejθ = cos θ + j sin θ, θ ∈ R,and the map :

[0, 2π] 3 θ 7→ ejθ ∈ Sp(X , σ)

is a 1−parameter group called the group of (global) gauge transformations.A quasi-free state ω on CCRWeyl(X , σ) is called gauge invariant if :

ω(W (x)) = ω(W (ejθx)), x ∈ X , θ ∈ R.

We can of course let ω act on CCRpol(X , σ). It is much more convenient then to use thecharged fields ψ(∗)(x) as generators of CCRpol(X , σ). We have then :

Proposition 5.5. A state ω on CCRpol(X , σ) is gauge invariant quasi-free iff :

ω (Πn1ψ∗(yi)Π

p1ψ(xi)) = 0, if n 6= p

ω (Πn1ψ∗(yi)Π

n1ψ(xi)) =

∑σ∈Sn

n∏i=1

ω(ψ∗(yi)ψ(xσ(i))).

It follows that a gauge invariant quasi-free state ω is uniquely determined by the ses-quilinear form

ω(ψ(y1)ψ∗(y2)) =: y1 · λ+y2.

Clearly λ+ ∈ Lh(Y ,Y∗). Let :ω(ψ∗(y2)ψ(y1)) =: y1 · λ−y2,


with λ− ∈ Lh(Y ,Y∗). From the commutation relations we have of course :

λ+ − λ− = q,

so λ− is determined by λ+, but nevertheless it is convenient to work with the pair (λ)±and to call λ± the complex covariances of ω.

The link between the real and complex covariances is as follows :

Lemma 5.6. We have :

η = Re(λ± ∓1

2q), η = λ± ∓

1

2q,

where η ∈ Lh(Y ,Y∗) is given by :

y1ηy2 := y1 ·ηy2 − iy1 ·ηjy2.

Note that the fact that η is sesquilinear follows from the gauge-invariance of ω. Fromthe above lemma one easily gets the following characterization of complex covariances.

Proposition 5.7. Let λ± ∈ Lh(Y ,Y∗). Then the following are equivalent :(1) λ± are the covariances of a gauge-invariant quasi-free state on CCRpol(Y , q),(2) λ± ≥ 0 and λ+ − λ− = q.

Proof. Since ω is gauge-invariant we have

j ∈ O(YR, η) ∩ Sp(YR,Reσ) = O(YR, η) ∩O(YR,Req).

From this fact and Lemma 5.6 we deduce that η ≥ 0 ⇔ λ+ ≥ 12q, and that the second

condition in Prop. 5.3 (with σ replaced by Reσ) is equivalent to

±q ≤ 2λ+ − q ⇔ λ± ≥ 0.

This completes the proof of the proposition. 2

5.4. Complexifying bosonic quasi-free states. If (X , σ) is real symplectic, we canform (CX , σC) which is charged symplectic. As real symplectic space it equals (X , σ) ⊕(X , σ). If ω is a quasi-free state on (X , σ) with real covariance η, we form a state ωCon ((CX )R,ReσC), with covariance ReηC, which is by definition gauge invariant. Hence,possibly after complexification, we can always reduce ourselves to gauge invariant quasi-free states.

5.5. Pure quasi-free states. In this subsection we discuss pure quasi-free states, whichturn out to be precisely vacuum states. To start the discussion, let us recall that fromProp. 5.3 (3) one has

(5.2) |x1·σx2| ≤ 2(x1·ηx1)12 (x2·ηx2)

12 , x1, x2 ∈ X .

We can complete the real vector space X w.r.t. the (semi-definite) symmetric form η, aftertaking the quotient by the vectors of zero norm as usual. Note that from (5.2) σ passesto quotient and to completion.

Denoting once again (X/Kerη)cpl by X , we end up with the following situation : (X , η)is a real Hilbert space, σ is a bounded, anti-symmetric form on X . However σ may verywell not be non-degenerate anymore, i.e. (X , σ) may just be pre-symplectic. One can show

14 C. GÉRARD

that if σ is degenerate, then the state on CCR(X , σ) with covariance η is not pure. In thesequel we hence assume that σ is non-degenerate on X . One can then prove the followingtheorem. The proof uses some more advanced tools, like the Araki-Woods representationand its properties.

Theorem 5.8. The state ω of covariance η is pure iff the pair (2η, σ) is Kähler, that isthere exists j ∈ Sp(X , σ) such that j2 = 1l and 2η = σj.

The link with Fock spaces and vacuum states is now as follows : if one equips X withthe complex structure j and the scalar product :

(x1|x2) := x1 · 2ηx2 + ix1 · ηjx2,

then Z := (X , (·|·)) is a complex Hilbert space. One can build the bosonic Fock spaceΓs(Z) and the Fock representation X 3 x 7→ eiφ(x) ∈ U(Γs(Z)). This representation isprecisely the GNS representation of the state ω, with GNS vector Ωω equal to the Fockvacuum.

For reference let us state the version of Thm. 5.8 using charged fields.

Theorem 5.9. Let λ± ∈ Lh(Y ,Y∗). Then the following are equivalent :(1) λ± are the covariances of a pure gauge-invariant quasi-free state on CCRpol(Y , σ),(2) there exists an involution κ ∈ U(Y , q) such that qκ ≥ 0 and λ± = 1

2q(κ± 1l).

(3) λ± ≥ ±12q, λ±q−1λ± = ±λ±, λ+ − λ− = q.

Thm. 5.9 can be easily deduced from Thm. 5.8.

5.6. Fermionic quasi-free states. We consider now an Euclidean space (X , ν). Withoutloss of generality we can assume that X is complete, i.e. (X , ν) is a real Hilbert space.We consider the CAR algebra CAR(X , ν), with the selfadjoint fermionic fields φ(x) asgenerators.

Definition 5.10. (1) a state ω on CARC∗(X , ν) is called quasi-free if

ω(φ(x1) · · ·φ(x2m−1)

)= 0,

ω(φ(x1) · · ·φ(x2m)

)=

∑σ∈Pair2m

sgn(σ)m∏j=1

ω(φ(xσ(2j−1))φ(xσ(2j))

),

for all x1, x2, · · · ∈ X , m ∈ N.(2) the anti-symmetric form β ∈ La(X ,X#) defined by :

x1 · βx2 := i−1ω([φ(x1), φ(x2)])

is called the covariance of the quasi-free state ω.

From the CAR it follows that

(5.3) ω(φ(x1)φ(x2)

)= x1·νx2 +

i

2x1·βx2, x1, x2 ∈ X .

Proposition 5.11. Let β ∈ La(X ,X#). Then the following are equivalent :(1) β is the covariance of a fermionic quasi-free state ω,(2) νC + i

2βC ≥ 0 on CX ,


(3) |x1·βx2| ≤ 2(x1·νx1)12 (x2·νx2)

12 , x1, x2 ∈ X .

Proof. as in the bosonic case (1) ⇒ (2) and (2) ⇔ (3) are easy to prove. The proof of(2) ⇒ (1) is more difficult, it relies on the Jordan-Wigner representation of CAR(X , ν)for X finite dimensional. 2

5.7. Gauge-invariant quasi-free states. We now assume that (X , ν) is equipped witha Kähler anti-involution j. This implies that ejθ ∈ U(X , ν) hence that there exists thegroup of global gauge transformations τθ, θ ∈ [0, 2π] defined by

τθφ(x) := φ(ejθx), x ∈ X .As in the bosonic case, a state ω on CAR(X , ν) is called gauge invariant if ω τθ = ω,∀θ ∈ [0, 2π]. Again it is more convenient to use the charged fields :

ψ(y) := φ(y) + iφ(jy), ψ∗(y) := φ(y)− iφ(jy),

as generators of CAR(Y). We recall that one sets q := 2νC ∈ Lh(Y ,Y∗), which is moreover positive definite.

Proposition 5.12. A state ω on CARC∗(Y) is gauge-invariant quasi-free iff :

ω (Πn1ψ∗(yi)Π

p1ψ(xi)) = 0, if n 6= p

ω (Πn1ψ∗(yi)Π

n1ψ(xi)) =

∑σ∈Sn

sgn(σ)n∏i=1

ω(ψ∗(yi)ψ(xσ(i))).

Again we can introduce the two complex covariances

ω(ψ(y1)ψ∗(y2)) =: y1 · λ+y2, ω(ψ∗(y2)ψ(y1)) =: y1 · λ−y2.

Proposition 5.13. Let λ± ∈ Lh(Y ,Y∗). Then the following are equivalent :(1) λ± are the covariances of a gauge-invariant quasi-free state on CAR(Y , q),(2) λ± ≥ 0 and λ+ + λ− = q.

5.8. Pure quasi-free states. We discuss pure quasi-free states in the fermionic case.We consider the general case, i.e. we do not assume the states to be gauge invariant.

Theorem 5.14. Let β ∈ La(X ,X#). Then β is the covariance of a pure quasi-free stateon CAR(X , ν) iff (ν, 1

2β) is Kähler, i.e. there exists j ∈ O(X , ν) such that j2 = −1l and

ν = 12βj.

We leave the formulation of the gauge-invariant version of this theorem as an exercise.

6. Lorentzian manifolds

6.1. Causality. Let M a smooth manifold of dimension n = d + 1. We assume that Mis equipped with a Lorentzian metric g, i.e. a smooth map

M 3 x 7→ gµν(x) ∈ Ls(TxM,T ∗xM),

with signature (−1, d). The inverse g−1(x) ∈ Ls(T∗xM,TxM) is traditionally denoted by

gµν(x). We set also |g|(x) := | det gµν(x)|

16 C. GÉRARD

Using the metric g we can define time-like, causal, etc. tangent vectors and vector fieldson M as on the Minkowski space-time. The set v ∈ TxM : vg(x)v = 0 is called thelightcone at x.M is then called time-orientable if there exists a global continuous time-like vector field

v. Once a time-orientation is chosen, one can define future/past directed time-like vectorfields.

One can similarly define time-like, causal etc. piecewise C1 curves, by requiring the saidproperty to hold for all its tangent vectors.

Definition 6.1. Let x ∈M . The causal, resp. time-like future, resp. past of x is the setof all y ∈ M that can be reached from x by a causal, resp. time-like future-, resp. past-directed curve, and is denoted J±(x), resp. I±(x). For U ⊂M , its causal, resp. time-likefuture, resp. past is defined as

J±(U) =⋃x∈U

J±(x), I±(U) =⋃x∈U

I±(x).

We define also the causal, resp. time-like shadow :

J(U) = J+(U) ∪ J−(U), I(U) = I+(U) ∪ I−(U).

The classification of tangent vectors in TxM can be naturally extended to linear sub-spaces of TxM .

Definition 6.2. A linear subspace E of TxM is space-like if it contains only space-likevectors, time-like if it contains both space-like and time-like vectors, and null (or light-like)if it is tangent to the lightcone at x.

If E ⊂ TxM we denote by E⊥ ⊂ TxM its orthogonal for g(x).

Lemma 6.3. A subspace E ⊂ TxM is space-like, resp. time-like, resp. null iff E⊥ istime-like, resp. space-like, resp. null

We refer to [F, Lemma 3.1.1] for the proof.

6.2. Globally hyperbolic manifolds.

Definition 6.4. A Cauchy hypersurface is a hypersurface Σ ⊂ M such that each inex-tensible time-like curve intersects Σ at exactly one point.

The following deep result is originally due to Geroch, with a stronger condition insteadof (1b), in this form it is due to Bernal-Sanchez.

Theorem 6.5. LetM be a connected Lorentzian manifold. The following are equivalent :(1) The following two conditions hold :

(1a) for any x, y ∈M , J+(x) ∩ J−(y) is compact,(1b) (causality condition) there are no closed causal curves.

(2) There exists a Cauchy hypersurface.(3) M is isometric to R×Σ with metric −βdt2+gt, where β is a smooth positive function,

gt is a Riemannian metric on Σ depending smoothly on t ∈ R, and each t × Σ isa smooth space-like Cauchy hypersurface in M .


Definition 6.6. A connected Lorentzian manifold satisfying the equivalent conditions ofthe above theorem is called globally hyperbolic.

The adjective ’hyperbolic’ comes from the connection with hyperbolic partial differen-tial equations : roughly speaking globally hyperbolic space-times are those on which theCauchy problem for the Klein-Gordon equation can be formulated and uniquely solved.

Definition 6.7. A function f : M → C is called space-compact if there exists K b Msuch that suppf ⊂ J(K). It is called future/past space-compact if there exists K b Msuch that suppf ⊂ J±(K). The spaces of such smooth functions are denoted by C∞sc (M),resp. C∞±sc(M).

7. Klein-Gordon fields on Lorentzian manifolds

7.1. The Klein-Gordon operator. Let (M, g) a Lorentzian manifold. TheKlein-Gordonoperator on M is :

P (x, ∂x) = −|g|−12∂µ|g|

12 gµν(x)∂ν + r(x),

acting on functions u : M → R. Here r ∈ C∞(M,R) represent a (variable) mass. Usingthe metric connection, one can write

P (x, ∂x) = −∇a∇a + r(x).

One can generalize the Klein-Gordon operator to sections of vector bundles, the mostimportant example being the bundle of 1−forms onM , which appears in the quantizationof Maxwell’s equation. One would like to interpret some space of smooth solutions of theKlein-Gordon equation :

(KG) P (x, ∂x)u = 0

as a symplectic space. Following the discussion in Sect. 5 we will consider complex solu-tions.

7.2. Conserved current.

Definition 7.1. We set for u1, u2 ∈ C∞(M) :

Ja(u1, u2) := u1.∇au2 −∇au1u2 ∈ C∞(M)

Ja(u1, u2) is a vector field, called a current in the physics literature. Using the rules tocompute with connections we obtain :

∇aJa(u1, u2) = ∇au1∇au2 + u1∇a∇au2 −∇a∇au1u2 −∇au1∇au2

= u1∇a∇au2 −∇a∇au1u2 = Pu1u2 − u1Pu2.

It follows that if Pui = 0 the vector field Ja(u1, u2) is divergence free. Moreover fromGauss formula we obtain :

Lemma 7.2 (Green’s formula). Let U ⊂M an open set with ∂U non characteristic. ThenˆU

u1Pu2 − Pu1u2dµg = −ˆ∂U

(u1∇au2 −∇au1u2

)nadσg.

18 C. GÉRARD

7.3. Advanced and retarded fundamental solutions. The treatment of the Klein-Gordon operator is quite similar to its Riemannian analog, i.e. the Laplace operator andstarts with the construction of fundamental solutions, i.e. inverses. An important differenceis the hyperbolic nature of the Klein-Gordon operator : to obtain a unique solution of theequation

Pu = v

say for v ∈ C∞0 (M), one has to impose extra support conditions on u, since there existsplenty of solutions of Pu = 0. The condition that (M, g) is globally hyperbolic is thenatural condition to be able to construct fundamental solutions. In the sequel we willassume that (M, g) is globally hyperbolic.

The main result is the following theorem, due to Leray.

Theorem 7.3. For any v ∈ C∞0 (M), there exist unique functions u± ∈ C∞±sc(M) thatsolve

P (x, ∂x)u± = v.

Moreover,

u±(x) = (E±v)(x) :=

ˆE±(x, y)v(y)dµg(y),

where E± ∈ D′(M ×M,L(V)) satisfy

P E± = E± P = 1l, suppE± ⊂

(x, y) : x ∈ J±(y).

Note that P is selfadjoint for the scalar product

(u|v) =

ˆM

uvdµg,

which by uniqueness implies that (E±)∗ = E∓. This also implies that by duality E± canbe applied to distributions of compact support.

Definition 7.4. E+, resp. E−, is called the retarded, resp. advanced Green’s function.

E := E+ − E−

is called the Pauli-Jordan function.

Note that E = −E∗, ie. E is anti-hermitian.

7.4. Cauchy problem. Once the existence of E± is established, it is easy to solve theCauchy problem (One can also go the other way around). Let us denote by Solsc(KG) thespace of smooth, space compact solutions of (KG). Let Σ be a smooth Cauchy hypersur-face. We denote by ρ : C∞sc (M)→ C∞0 (Σ)⊕ C∞0 (Σ) the map :

ρu := (ρ0u, ρ1u) = (u|Σ, nµ∂µu)|Σ).

Theorem 7.5. Let f ∈ C∞0 (Σ) ⊗ C2. Then there exists a unique u ∈ Solsc(KG) suchthat ρu = f . It satisfies suppu ⊂ J(suppf0 ∪ suppf1) and is given by

(7.1) u(x) = −ˆ

Σ

nµ∇yµE(x, y)f0(y)dσ(y) +

ˆΣ

E(x, y)f1(y)dσ(y).

We will set u =: Uf , for u given by (7.1).


Remark 7.6. Let us denote by ρ∗i : D′(Σ)→ D′(M) the adjoints of ρi. Let us also set

qΣ =

(0 1l−1l 0

)∈ L(C∞0 (Σ)⊕ C∞0 (Σ)).

Then (7.1) can be rewritten as :

(7.2) 1l = Eρ∗qΣρ, on Solsc(KG),

or equivalently

(7.3) U = E ρ∗ qΣ, on C∞0 (Σ)⊕ C∞0 (Σ).

Proof. We will just prove that the solution u of the Cauchy problem is given by theabove formula. We fix f ∈ C∞0 (M) and apply Green’s formula to u1 = E∓f , u1 = u, andU = J±(Σ). We obtain :´

J+(Σ)fudµg =

´Σ

(E−f∇au−∇aE−fu

)nadσg,´

J−(Σ)fudµg =

´Σ−(E+f∇au−∇aE+fu

)nadσg,

Summing these two identities we obtain :ˆM

fudµg =

ˆΣ

(−Ef∇au+∇aEfu

)dσg

To complete the proof of the formula, it suffices to introduce the distribution kernel E(x, y)to express Ef as an integral and to use that E = −E∗. Details are left to the reader. 2

7.5. Symplectic structure of the space of solutions. By Thm. 7.5 we know that :

ρ : Solsc(KG)→ C∞0 (Σ)⊕ C∞0 (Σ),

is bijective, with inverse U . For u, v ∈ Solsc(KG) with ρu =: f , ρv =: g we set :

(7.4)u1 · σu2 :=

´Σf 1g0 − f 0g1dσg =

´ΣJa(u, v)nadσg

fσΣg :=´

Σf 1g0 − f 0g1dσg.

It is obvious from Thm. 7.5 that (Solsc(KG), σ) is a (complex) symplectic space. FromGauss formula we know that σ is independent on the choice of the Cauchy surface Σ.

7.6. Space-time fields.

Theorem 7.7. (1) consider E : C∞0 (M)→ C∞(M). Then RanE = Solsc(KG), KerE =PC∞0 (M).

(2) One hasEf 1 · σEf2 = −(f1|Ef2), fi ∈ C∞0 (M).

Part 1) of the theorem can be nicely rephrased by saying that the sequence :

0 −→ C∞0 (M)P−→C∞0 (M)

E−→ Solsc(KG)P−→ 0

is exact.Proof. 1) : since PE± = 1l, we have PE = 0, and taking adjoints also EP = 0.This shows that EC∞0 (M) ⊂ Solsc(KG) and PC∞0 (M) ⊂ KerE. It remains to prove theconverse inclusions :

20 C. GÉRARD

1a) Solsc(KG) ⊂ EC∞0 (M) : let u ∈ Solsc(KG). Since u is space-compact, we can findcutoff functions χ± ∈ C∞±sc(M) such that χ+ + χ− = 1 on suppu. We have some compactsets K ⊂ K1 ⊂ K2 such that suppu ⊂ J(K), suppχ± ⊂ J±(K2), and χ± = 1 on J±(K1).It follows that supp∇χ± ⊂ J±(K2)\J±(K2), hence supp∇χ±∩ suppu ⊂ J∓(K2)∩J±(K).This set is compact by the global hyperbolicity of M .

We set now u± = χ±u, f = Pu+ = −Pu−. By the above discussion f ∈ C∞0 (M) henceu± = ±E±f and u = u+ + u− = Ef .

1b) KerE = PC∞0 (M) : let u ∈ C∞0 (M) such that Eu = 0 and let f = E±u. Thensuppf ⊂ J+(suppu) ∩ J−(suppu), hence again by global hyperbolicity, f ∈ C∞0 (M).

2) : from (7.2) we obtain :

E = −Eρ∗qΣρE = (ρ E)∗ qΣ (ρ E).

This implies (2). 2

We now summarize the discussion as follows :

Theorem 7.8. (1) the following spaces are symplectic spaces :

(C∞0 (M)/PC∞0 (M),−E), (Solsc(KG), σ), (C∞0 (Σ)⊕ C∞0 (Σ), σΣ).

(2) the following maps are symplectomorphisms :

(C∞0 (M))/PC∞0 (M),−E)E−→(Solsc(KG), σ)

ρ−→(C∞0 (Σ)⊕ C∞0 (Σ), σΣ).

7.7. Quasi-free states for the free Klein-Gordon field. We can now consider quasi-free states on any of the symplectic spaces in Thm. 7.8. The most natural one is

(C∞0 (M))/PC∞0 (M), E)

which leads to space-time fields. The associated CCR algebra will be denoted simply byCCR(C∞0 (M), E), ignoring the need to pass to quotient to get a true symplectic space.

Strictly speaking we would write symbols like

φ([f ]), [f ] ∈ C∞0 (M)/PC∞0 (M).

We write this asφ(f)” = ”

ˆM

φ(x)f(x)dµg” = ”〈φ, f〉

if Pφ(x) = 0, i.e. the quantum field φ satisfies the KG equation.causality : denoting by φ(f) the (selfadjoint) fields associated to [f ] ∈ C∞0 (M))/PC∞0 (M),

we have[φ(f), φ(g)] = Re(f |Eg) = 0,

if suppf , suppg are causally disjoint. This follows from the fact that suppE(x, y) ⊂(x, y) ∈M ×M : x ∈ J(y).

Let us now consider a gauge-invariant quasi-free state ω, defined by the complex co-variances (λ±). Recall that λ± are sesquilinear forms on C∞0 (M))/PC∞0 (M). One mayassume that they are obtained from sesquilinear forms Λ± on C∞0 (M) which pass toquotient, i.e. such that :

(7.5) P ∗ Λ± = Λ± P = 0.


It is also natural to assume that Λ± are continuous sesquilinear forms, hence by theSchwartz kernel theorem have distributional kernels Λ±(x, y) ∈ D′(M ×M). Then (7.5)becomes, using that P = P ∗ :

(7.6) P (x, ∂x)Λ±(x, y) = P (y, ∂y)Λ±(x, y) = 0.

8. Free Dirac fields on Lorentzian manifolds

8.1. The Dirac operator. Let (M, g) be a Lorentzian manifold. To define correctlya Dirac operator we need a spin structure. We keep the discussion simple, and use aframework in [DG], assuming that the manifold M is parallelizable.

Let V be a complex, finite dimensional vector space. We assume that there exists a mapM 3 x 7→ γa(x) ∈ L(V) such that

[γa(x), γb(x)]+ = 2gab(x)

Of course to make this definition clean one should use the language of bundles. One canthink of γa as x dependent Dirac matrices.

Definition 8.1. Let M 3 x 7→ m(x) ∈ L(V). The operator on C∞(M ;V)

D := γa∂xa +m(x)

is called a Dirac operator on M . The Dirac equation is :

(D) Dζ = 0.

We need some more structure to be able to quantize the Dirac equation. We assumethat there exists a smooth map :

(8.7) M 3 x 7→ λ(x) ∈ Lh(V ,V∗)

such that :

(8.8)γa(x) is selfadjoint for λ(x),

m(x)− 12∇aγ

a(x) is anti-selfadjoint for λ(x).

We equip C∞0 (M ;V) of the sesquilinear form

(ζ1|ζ2) =

ˆM

ζ1(x) · λ(x)ζ2(x)dµg,

and we obtain that if (8.8) holds then D∗ = −D.

8.2. Conserved current. We set for ζ1, ζ2 ∈ C∞(M ;V)

Ja(ζ1, ζ2) := ζ1(x) · λ(x)γa(x)ζ2(x).

One proves that if ζi are solutions of the Dirac equation, then

∇aJa(ζ1, ζ2) = 0.

We also get the Green’s formula :

22 C. GÉRARD

Lemma 8.2. Let U ⊂M an open set with ∂U non characteristic. ThenˆU

ζ1 · λDζ2 − Dζ1 · λζ2dµg = −ˆ∂U

ζ1 · λγaζ2nadσg,

for γa = gabγb.

8.3. Advanced and retarded fundamental solutions. We assume now that (M, g) isglobally hyperbolic. It is easy to construct fundamental solutions for the Dirac operator :in fact DD is a Klein-Gordon operator, acting on vector valued functions, but with scalarprincipal part equal to −∇a∇a. The existence of fundamental solutions E± for D2 yieldsthe fundamental solutions S± = DE±. This is summarized in the next theorem.

Theorem 8.3. For any f ∈ C∞0 (M ;V), there exist unique functions ζ± ∈ C∞±sc(M ;V)that solve

Dζ± = f.

Moreover,

ζ±(x) = (S±f)(x) :=

ˆS±(x, y)fvy)dµg(y),

where S± ∈ D′(M ×M,L(V)) satisfy

D S± = S± D = 1l, suppS± ⊂

(x, y) : x ∈ J±(y).

If (8.8) holds then

λ(x)S±(x, y) = −S∓(x, y)∗λ(y),

i.e. S±∗ = −S∓, if we equip C∞0 (M ;V) with the (non-positive) scalar product obtainedfrom λ.

Definition 8.4. S± are called the retarded/advanced Green’s functions.

S := S+ − S−

is called the Pauli-Jordan function.

Note that S = S∗, i.e. S is hermitian.

8.4. Cauchy problem. We state without proof the existence and uniqueness result forthe Cauchy problem. We denote by ρ : C∞sc (M ;V) → C∞0 (Σ;V) the trace on Σ and bySolsc(D) the space of smooth space-compact solutions of the Dirac equation.

Theorem 8.5. Let Σ be a Cauchy surface. Then for any f ∈ C∞0 (Σ;V) there exists aunique ζ ∈ Solsc(D) such that ρζ = f . It satisfies suppζ ⊂ J(suppf) and is given by :

ζ(x) = −ˆ

Σ

S(x, y)γa(y)na(y)f(y)dµg(y).


8.5. Hermitian structure on the space of solutions. We would like to equip Solsc(D)with a (positive) hermitian structure. To do this we need an additional positivity condi-tion. We assume that there exists a global, time-like future directed vector field v suchthat

(8.9) λ(x)γa(x)va(x) > 0, ∀x ∈M.

It can be shown that if this is true for one such vector field, it is automatically true forall such vector fields, in particular for the normal vector to a given Cauchy surface. Forζ1, ζ2 ∈ Solsc(D) with ρζi = fi we set :

(8.10)ζ1 · νζ2 :=

´ΣJa(ζ1, ζ2)nadσg =

´Σf 1 · λ(y)γa(y)f2(y)na(y)dσg(y),

f 1 · νΣf2 =´

Σf 1 · λ(y)γa(y)f2(y)na(y)dσg(y).

From Thm. 8.5 we see that (Solsc(D), ν) is a hermitian space, and from (8.9) ν is positivedefinite. Moreover from Gauss formula ν is independent on the choice of a Cauchy surface.

8.6. Space-time fields.

Theorem 8.6. (1) consider S : D(M ;V) → C∞(M ;V). Then RanS = Solsc(D) andKerS = DC∞0 (M ;V).

(2) One has :Sf 1 · νSf2 = (f1|Sf2), fi ∈ C∞0 (M ;V).

Proof. 1) is left to the reader.2) : Thm. 8.5 can be rewritten as

1l = Sρ∗γa∗naρ, on Solsc(D),

hence by 1)S = Sρ∗γa∗naρS,

which implies 2) since S∗ = S. 2

Theorem 8.7. (1) the following spaces are pre-Hilbert spaces :

(C∞0 (M ;V)/DC∞0 (M ;V), S), (Solsc(D), ν), (C∞0 (Σ;V), νΣ).

(2) the following maps are unitary :

(C∞0 (M ;V)/DC∞0 (M ;V), S)S−→(Solsc(D), ν)

ρ−→(C∞0 (Σ;V), νΣ).

8.7. Quasi-free states for the free Dirac field. As for the Klein-Gordon case wechoose the pre-Hilbert space :

(C∞0 (M ;V)/DC∞0 (M ;V), S).

The associated CAR algebra is denoted by CAR(C∞0 (M ;V ), S).The causality is a bit different : we obtain

[φ(f), φ(g)]+ = Re(f |Sg) = 0,

if suppf , suppg are causally disjoint, i.e. fields supported in causally disjoint regions anti-commute. This puzzle is solved by considering only even elements of CAR(C∞0 (M ;V)/DC∞0 (M ;V), S)as true physical observables.

24 C. GÉRARD

Let us consider a gauge-invariant quasi-free state ω, defined by the complex covariances(λ±). Recall that λ± are sesquilinear forms on C∞0 (M ;V)/DC∞0 (M ;V)). Again one as-sumes that they are obtained from sesquilinear forms Λ± on C∞0 (M ;V) which pass toquotient, i.e. such that :

(8.11) D Λ± = Λ± D = 0.

Introducing as before the distributional kernels Λ±(x, y) ∈ D′(M ×M) ⊗ V ⊗ V∗. Then(8.11) becomes :

(8.12) D(x, ∂x)Λ±(x, y) = D(y, ∂y)Λ±(x, y) = 0.

9. Microlocal analysis of Klein-Gordon quasi-free states

9.1. The need for renormalization. The stress -energy tensor for a classical Klein-Gordon field is given by :

Tµν(x) = ∇µφ(x)∇νφ(x)− 1

2gµν(x)

(gab(x)∇aφ(x)∇bφ(x)−m2φ2(x)

).

For a quantized Klein-Gordon field one would like to be able to define Tµν(x) as anoperator valued distribution. This means the following :

we choose a state ω (say a quasi-free state), and fix f1, . . . , fn, g1, . . . , gp ∈ C∞0 (M).Then

x 7→ ω(n∏1

φ(fi)Tµν(x)

p∏1

φ(gj))

should be a distribution on M . This is never the case, even on Minkowski space : forexample ω(φ2(x)) should be the trace on x = x′ of ω(φ(x)φ(x′)) = ω2(x, x′), which makesno sense. We need to subtract the singular part of ω2(x, x′) near the diagonal, i.e. toperform a renormalization.

Another requirement is that the renormalization procedure should be ’covariant’ : itshould depend only on the metric in a arbitrarily small neighborhood of x. This impliesthat is will be covariant under isometric embeddings.

The procedure is as follows. One first performs the ’point-splitting’, i.e. consider

T µν(x, y) = ∇µφ(x)∇νφ(y)− 1

2gµν(x)

(gab(x)∇aφ(x)∇bφ(y)−m2φ(x)φ(y)

).

One then remove the singular part by setting :

:φ(x)φ(y) : = φ(x)φ(y)− cHad(x, y)1l,

where cHad(x, y) is a well chosen distributional kernel. Then one has to check that thedistributions

ω(n∏1

φ(xi)Tµν(x, y) :

p∏1

φ(yj))

have well-defined restrictions to x = y.


9.2. Old form of Hadamard states. Choose a Cauchy surface Σ. A causal normalneighborhood N of Σ in M is an open neighborhood of Σ such that Σ is a Cauchy surfaceof (N, g) and for each x, x′ ∈ N such that x ∈ J+(x′) there exists a convex normal openset containing J+(x′) ∩ J−(x). We fix a time function T : M → R , i.e.a smooth functionwhich increases towards the future, and a open neighborhood O ⊂ M ×M of the set ofpairs of causally related points (x, x′) such that J±(x)∩J∓(x′) are contained in a convex,normal open neighborhood.

One fixes also O′ an open neighborhood in N ×N of the set of pairs of causally relatedpoints such that O′ ⊂ O.

The squared geodesic distance σ(x, x′) is smooth on O and well defined on the subsetof N ×N consisting of causally related points. The van Vleck-Morette determinant is

∆(x, x′) := − det(−∇α∇β′σ(x, x′))|g|−12 (x)|g|−

12 (x′).

One can then construct a sequence of functions v(n) ∈ C∞(O ×O) of the form

v(n)(x, x′) =n∑1

σ(x, x′)ivi(x, x′),

where the vi ∈ C∞(O × O) are uniquely determined by some transport equations (theso-called Hadamard recursion relations).

One then defines a sequence of distributions c(n)Had ∈ D′(O ×O) for n ≥ 1 by :

c(n)Had(x, x′) = limε→0+

(2π)2∆(x, x′)12

σ(x, x′) + 2iε(T (x)− T (x′)) + ε2

+ limε→0+ v(n)(x, x′) ln (σ(x, x′) + 2iε(T (x)− T (x′))) .

Let us now give the old definition of Hadamard states, restricting ourselves to real Klein-Gordon fields.

Definition 9.1. A quasi-free state ω on CCR(C∞0 (M), E) is a Hadamard state if for anym ∈ N there exists n ∈ N such that ω2 − c(n)

Had is of class Cm in O′.

9.3. The wavefront set on a manifold. Let M be a manifold, T ∗M be the cotangentbundle. The zero section T ∗0M will be denoted by Z.

We recall the spaces : D(M) (smooth compactly supported functions), D′(M) (distri-butions), E(M) (smooth functions with well-known topology), E ′(M) (distributions withcompact support).

9.4. Operations on conic sets. A set Γ ⊂ T ∗M\Z is conic if (x, ξ) ∈ Γ ⇒ (x, tξ) ∈ Γfor all t > 0. Let Γi ⊂ T ∗M\Z, i = 1, 2 be conic sets. We set :

−Γ := (x,−ξ) : (x, ξ) ∈ Γ,Γ1 ⊕ Γ2 := (x, ξ1 + ξ2) : (x, ξi) ∈ Γi.

Let Mi, i = 1, 2 be two manifolds and Γ ⊂ T ∗M1×M2\Z be a conic set. The elements ofT ∗M1 ×M2\Z will be denoted by (x1, ξ1, x2, ξ2) which allows to consider Γ as a relation

26 C. GÉRARD

between T ∗M2 and T ∗M1, still denoted by Γ. Clearly Γ maps conic sets into conic sets.We set :

Γ′ := (x1, ξ1, x2,−ξ2) : (x1, ξ1, x2, ξ2) ∈ Γ,

M1Γ := (x1, ξ1) : ∃x2 such that (x1, ξ1, x2, 0) ∈ Γ = Γ(Z2),

ΓM2 := (x2, ξ2) : ∃x1 such that (x1, 0, x2, ξ2) ∈ Γ = Γ−1(Z1).

9.5. Operations on distributions.(1) Complex conjugation : if u ∈ D′(M) then WFu = −WF (u).(2) Tensor product : if ui ∈ D′(Mi), i = 1, 2 then

WF (u1 ⊗ u2)

⊂ (WF (u1)×WF (u2)) ∪ (suppu1 × 0)×WF (u2) ∪WF (u1)× (suppu2 × 0)⊂ (WF (u1)×WF (u2)) ∪ Z1 ×WF (u2) ∪WF (u1)× Z2.

(3) Restriction to a submanifold : let S ⊂ M a submanifold. The conormal bundleT ∗SM is defined as :

T ∗SM := (x, ξ) ∈ T ∗M\Z : x ∈ S, : ξ · v = 0 ∀v ∈ TxS.

If u ∈ D′(M) the restriction u|S of u to S is well defined if WFu ∩ T ∗SM = ∅. Onehas

WFu|S ⊂ (x, ξ|TxS) : x ∈ S, (x, ξ) ∈ WF (u).

(4) Product : if ui ∈ D′(M), i = 1, 2 then u1u2 is well defined ifWF (u1)⊕WF (u2)∩Z =∅ and then :

WF (u1u2) ⊂ WF (u1) ∪WF (u2) ∪WF (u1)⊕WF (u2).

(5) Kernels : let K : D(M2)→ D′(M1) be linear continuous and K(x1, x2) ∈ D′(M1 ×M2) its distributional kernel.

then Ku is well defined for u ∈ E ′(M2) if WF (u) ∩WF ′M2(K) = ∅ and then :

WF (Ku) ⊂ M1WF (K) ∪WF ′(K) WF (u).

(6) Composition : let K1 ∈ D′(M1 ×M2), K2 ∈ D′(M2 ×M3), where K2 is properlysupported ie the projection : suppK2 →M2 is proper. Then K1 K2 is well definedif :

WF ′(K1)M2 ∩ M2WF (K2) = ∅,

and then

WF ′(K1 K2) ⊂ WF ′(K1) WF ′(K2) ∪ (M1WF (K1)× Z3) ∪ (Z1 ×WF (K2)M3).


9.6. Parametrices for the Klein-Gordon operator. Once we choose an orientation,we can define for x ∈ M the open future/past light cones V ±x ⊂ TxM . We denote byV ∗x± ⊂ T ∗xM the dual cones

V ∗x± = ξ ∈ T ∗xM : ξ · v > 0, ∀v ∈ Vx±, v 6= 0.

For simplicity we write

ξ 0 ( resp. ξ 0) if ξ ∈ V ∗x+ ( resp. V ∗x−).

ξ 0 ( resp. ξ 0) if ξ ∈ (V ∗x+)cl ( resp. (V ∗x−)cl).

The principal symbol of the Klein-Gordon operator P is p(x, ξ) = ξµgµν(x)ξν and one sets

N := p−1(0) ∩ T ∗M\Z,

called the characteristic manifold of P . Note that N splits into its two connected compo-nents (positive/negative energy shells) :

N = N+ ∪N−, N± = N ∩ ±ξ 0.

We denote by Hp the Hamilton field of p. We denote by X = (x, ξ) the points in T ∗M\Z.The bicharacteristic (Hamilton curve for p) passing through X is denoted by B(X) ForX, Y ∈ N we write X ∼ Y if Y ∈ B(X). Clearly this is an equivalence relation.

For X ∼ Y , we write X Y (resp. X ≺ Y ) if X comes strictly after (before) Y w.r.t.the natural parameter on the bicharacteristic curve through X and Y .

We recall Hörmander’s propagation of singularities theorem :

Theorem 9.2. Let u ∈ D′(M) such that Pu ∈ C∞(M). Then

WF (u) ⊂ N, X ∈ WF (u)⇒ B(X) ⊂ WF (u).

The bicharacteristic relation of P is the set :

C := (X, Y ) ∈ N ×N : X ∼ Y .

We set :∆N := (X,X) ∩N ×N

the diagonal in N ×N .Parametrices (i.e. inverses modulo smoothing operators) of operators of real principal

type (of which Klein-Gordon operators are an example) were studied by Duistermaatand Hörmander in the famous paper [DH]. They introduced the notion of distinguishedparametrices, i.e. parametrices which are uniquely determined (modulo smoothing termsof course), by the wavefront set of their kernels. Distinguished parametrices are in one toone correspondence with orientations of C, defined below.

Definition 9.3. An orientation of C is a partition of C\∆N as C1 ∪ C2 where Ci areopen sets in C\∆N and inverse relations (ie Exch(C1) = C2).

Note that Ci 6= ∅, Ci 6= C\∆N (because they are inverse relations) and Ci are openand closed in C\∆N . Therefore Ci are union of connected components of C\∆N .

28 C. GÉRARD

Theorem 9.4 (D-H). Let C\∆N = C1 ∪ C2 an orientation of C. Then there existsparametrices Ei, i = 1, 2 of P such that :

WF ′(Ei) ⊂ ∆∗ ∪ Ci,

where ∆∗ is the diagonal in T ∗M\Z×T ∗M\Z. Any left or right parametrix with the sameproperty is equal to E1 or E2 modulo C∞.

Orientations of C are themselves in one to one correspondence to the partitions ofN1 ∪ N2 = N into open and closed sets, ie into connected components of N . For theKlein-Gordon operator N has two connected components :

N± := X ∈ N : ξ ∈ V ∗x±,

which are invariant under the bicharacteristic flow, hence two orientations, hence fourdistinguished parametrices. The two connected components are N+, N−. The two orien-tations are

C\∆N = C+ ∪ C− for

C+ := (X, Y ) ∈ C : x ∈ J+(y), C− := (X, Y ) ∈ C : x ∈ J−(y)

andC\∆N = C+ ∪ C− for

C+ := (X, Y ) ∈ C : X ≺ Y , C− := (X, Y ) ∈ C : X Y .Note that :

X ≺ Y ⇔ X ∼ Y andx ∈ J+(y), X, Y ∈ N+,or x ∈ J−(y), X, Y ∈ N−;

X Y ⇔ X ∼ Y andx ∈ J−(y), X, Y ∈ N+,or x ∈ J+(y), X, Y ∈ N−.

The parametrices associated to these orientations are well-known in physics, we havealready encountered two of them, namely E±.

Feynman : denoted EF :WF(EF )′ = ∆∗ ∪ C+

anti-Feynman : denoted EF :WF(EF )′ = ∆∗ ∪ C−

retarded : E+ :WF(E+)′ = ∆∗ ∪ C+,

advanced : E− :WF(E−)′ = ∆∗ ∪ C−.

The parametrices E± are more fundamental since they are used to define the symplec-tic form E. The parametrices EF , EF appear in connection with the vacuum state onMinkowski space, and with Hadamard states on general curved spacetimes.


9.7. Examples. Assume that P = ∂2t + ε2 (whatever ε2 is, for example a real number).

Then :E+(t) = θ(t) sin εt

ε,

E−(t) = −θ(−t) sin εtε,

EF (t) = 12iε

(eitεθ(t) + e−itεθ(−t)

),

EF (t) = − 12iε

(e−itεθ(t) + eitεθ(−t)

),

for θ(t) =Heaviside function.We prove some properties of the distinguished parametrices.

Lemma 9.5. We have :1) WF ′(E+ − E−) = C,

2) WF ′(EF − E+) = C ∩N− ×N−,3) WF ′(EF − E−) = C ∩N+ ×N+.

Proof.1) : since E+ and E− have disjoint wave front sets above x1 6= x2, we see that abovex1 6= x2

WF ′(E+ − E−) = WF ′(E+) ∪WF ′(E−) = C\∆N .

Since P1(E+−E−) ∈ C∞ by propagation of singularities we obtain that ∆N ⊂ WF ′(E+−E−). This proves 1).

2) : above (x1, x2) : x1 ∈ J−(x2) we have

WF ′(EF − E+) = WF ′(EF ) = (X1, X2) : x1 ∈ J−(x2), ξ1 0.Using again that P1(E+ − EF ) ∈ C∞ we obtain that WF ′(EF − E+) = C ∩ N− × N−.The proof of 3) is similar. 2

9.8. The theorem of Radzikowski.

Definition 9.6. Let Λ± : D(M)→ E(M) be linear continuous. The pair Λ± satisfies theHadamard condition if :

(Had) WF(Λ±)′ = (X1, X2) ∈ N± ×N± : X1 ∼ X2.The pair Λ± satisfies the generalized Hadamard condition if there exists conic sets Γ±with (X1, X2) ∈ Γ± ⇒ ±ξ1 0,±ξ2 0 such that

(genHad) WF (Λ±)′ ⊂ Γ±.

We introduce the following conditions on a pair Λ± :

(KG) P Λ±,Λ± P smoothing,

(CCR) Λ+ − Λ− − iE smoothingThe following theorem is the theorem of Radzikowski (extended to the complex case).

Theorem 9.7. The following three conditions are equivalent :(1) Λ± satisfy (Had), (KG) and (CCR),(2) Λ± satisfy (genHad), (KG) and (CCR),

30 C. GÉRARD

(3) one has :Λ± = i(EF − E∓) modulo C∞(M ×M).

Proof. (1)⇒ (2) : obvious.(2)⇒ (3) : set S± = i(EF −E∓) By Lemma 9.5 we have WF(S±)′ ⊂ C∩(N±×N±). By

(KG) and Thm. 9.2 we obtain that WF(Λ±)′ ⊂ N ×N . Using then (genHad) we obtainthat WF(Λ±) ⊂ Γ± ∩N ×N ⊂ N+ ×N+. This implies that

WF(Λ± − S±)′ ⊂ N± ×N±,which implies in particular that

(9.1) WF(Λ+ − S+)′ ∩WF(Λ− − S−)′ = ∅.By (CCR) we have also

(Λ+ − S+)− (Λ− − S−) = iE − (S+ − S−) = iE − iE mod C∞(M ×M).

By (9.1) this implies that both Λ± − S± are smooth.(3)⇒ (1) : (KG) and (CCR) are obvious, (Had) follows from Lemma 9.5. 2

There is a remaining painful step, which I will only briefly explain. I will consider realfields for simplicity. The conclusion of the above theorem for the real covariance ω2(x, x′)is that ω2 = i(EF − E+) mod C∞. Then one has to perform some painful computationsto prove that i(EF − E+) = cHad mod C∞.

9.9. Return to the stress-energy tensor. Let us forget the derivatives and juste consi-der :φ2(x) :. We put just one field on each side (extension to several fields is straightfor-ward) :

We compute the expectation value of

φ(x1)φ(x)φ(y)φ(x2)− φ(x1)φ(x2)cHad(x, x′),

in the state ω. By the quasi-free property, we obtain a sum of two types of terms :

ω2(x1, x)ω2(x′, x2), ω2(x, x2)ω2(x1, x′),

andω2(x1, x2)× (ω2(x, x′)− cHad(x, x′)).

The second term has a restriction to the diagonal x = x′, since ω2 − cHad is smooth.For the first term we consider the wavefront set of the distribution. By the tensor

product rule we obtain :

WF(ω2)×WF(ω2) ∪ (suppω2 × 0)×WF(ω2) ∪WF(ω2)× (suppω2 × 0).We compute the conormal to S = x = x′ :

T ∗SM = (x1, ξ1, x, ξ, y, η, x2, ξ2) : ξ1 = ξ2 = 0, x = y, ξ = −ηWe have to show the intersection is empty, which is obvious since (X, Y ) ∈ WF (ω2)implies ξ, η 6= 0.

10. Construction of Hadamard states

It is not a priori obvious that Hadamard states exist on an arbitrary globally hyperbolicspacetime. In this section we explain some constructions of Hadamard states.


10.1. Ultrastatic spacetimes. Consider an ultrastatic spacetime (R×Σ, g), g = −dt2 +

h. Let a = −∆h +m2 on Σ and ε = a12 . One can then define the vacuum state ωvac, whose

two-point function is given by the kernel :

Λvac± (t) =

1

2εe±itε,

i.e. writing u ∈ C∞(R× Σ) as R 3 t 7→ u(t) ∈ C∞(Σ) :

Λvac± u(t) =

ˆR

1

2εe±i(t−s)εu(s)ds.

Sahlmann and Verch [SV] have shown that ωvac is a Hadamard state. Another proof canbe given by using the arguments of Subsect. 10.3.

Similarly one can define the thermal state at temperature β−1, β > 0 with kernel

Λβ±(t) :=

1

2ε(1− e−βε)(e±itε + e∓itεe−βε).

It is again a Hadamard state since e−βε is a smoothing operatorThe conclusion is that vacuum or thermal states on ultrastatic (or static) spacetimes

are Hadamard states.

10.2. The FNW deformation argument. Let (M, g) a globally hyperbolic space-time.The deformation argument of Fulling, Narcowich and Wald [FNW] is based on two facts.

The first fact is the so-called time slice property : if U ⊂ M is a neighborhood of aCauchy surface Σ, then for any u ∈ C∞0 (M) there exists v ∈ C∞0 (U) such that u − v ∈PC∞0 (M). In other words

C∞0 (M)/PC∞0 (M) = C∞0 (U)/PC∞0 (M).

This implies that a pair of distributions Λ± ∈ C∞0 (U × U) satisfying :

P Λ± = Λ± P = 0,

Λ+ − Λ− = −iE on U × U,Λ± ≥ 0 on C∞0 (U)

generate a quasi-free state on M .The second fact is Hörmander’s propagation of singularities theorem (Thm. 9.2) : If Λ±

satisfy (Had) (or (genHad)) over U×U , and P Λ± = Λ± P = 0, then Λ± satisfy (Had)or (genHad) globally.

The conclusion of these two facts is that if g1, g2 are two Lorentzian metrics suchthat (M, gi) is globally hyperbolic, have a common Cauchy surface Σ and coincide in aneighborhood of Σ, then a Hadamard state for the Klein-Gordon field on (M, g1) generatesa Hadamard state for the Klein-Gordon field on (M, g2).

One argues then as follows : let us fix a Cauchy surface Σ for (M, g) and identify (M, g)with (R × Σ,−c2(t, x)dt2 + hij(t, x)dxidxj). We set Σt = t × Σ. We fix a real functionr ∈ C∞(M) and consider P = −∇a∇a + r(x) — the associated Klein-Gordon operator.One chooses an ultra-static metric

gus = −dt2 + hjk,us(x)dxjdxk, rus(x) = m2 > 0,

32 C. GÉRARD

and an interpolating metric gint = −c2int(t, x)dt2 +hjk,int(x)dxjdxk, and real function rint ∈

C∞(M) such that (gint, rint) = (g, r) near ΣT , (gint, rint) = (gus,m2) Σ−T .

The vacuum state ωvac for (M, gus) is Hadamard for Pus, hence generates a Hadamardstate for Pint, which itself generates a Hadamard state ω for P .

Using the Cauchy evolution operator from Σ−T to ΣT , one sees that ω is pure, sinceωvac is pure.

10.3. Construction of Hadamard states by pseudodifferential calculus. Let usnow briefly describe another construction given in a joint work with Michal Wrochna[GW1]. It relies on the choice of a Cauchy surface Σ and on the global pseudodifferentialcalculus on Σ.

We identify M with R×Σ, with the split metric g = −c2(t, x)dt2 +hij(t, x)dxidxj, andwe set Σs := s × Σ ⊂M . For simplicity we will assume that Σ is either equal to Rd orto a compact manifolds, but much more general situations can be treated as well.

We denote by Ψm(Σ) the space of (uniform) pseudodifferential operators of order m onΣ, corresponding to quantization of symbols in Sm1,0(T ∗Σ), and by Ψph(Σ) the subspace ofpseudodifferential operators with poly-homogeneous symbols. We set

Ψ∞(Σ) =⋃m∈R

Ψm(Σ), Ψ−∞(Σ) =⋂m∈R

Ψm(Σ).

We denote by Hs(Σ) the Sobolev space of order s on Σ and

H∞(Σ) =⋂s∈R

Hs(Σ), H−∞(Σ) =⋃s∈R

Hs(Σ).

We use the third version of the phase space, namely (C∞0 (Σ)⊗ C2, q), where we recallthat q = iσΣ. We embed C∞0 (Σ) ⊗ C2 into D′(Σ) ⊗ C2 using the natural scalar producton C∞0 (Σ)⊗ C2, i.e. we identify sesquilinear forms with operators.

Quasi-free states are now defined by a pair of covariances λ± on C∞0 (Σ)⊗ C2, and therelationship with the space-time covariances Λ± is :

Λ± = (ρ E)∗ λ± (ρ E).

It is natural to restrict attention to states with covariances λ± ∈ Ψ∞(Σ) ⊗ M(C2). Itcan be shown that if a state has pseudodifferential covariances on Σs for some s it haspseudodifferential covariances on Σs for any s.

The most important part is to characterize the Hadamard condition in terms of λ±,which relies on the construction of a parametrix for the Cauchy problem on Σ, see Prop.10.3 below.

10.3.1. The model Klein-Gordon equation. By a change of coordinates and conjugationwith a convenient weight, one can reduces oneself to the equation :

∂2t u+ a(t, x, ∂x)u = 0,

where a(t, x, ∂x) is a second order, elliptic selfadjoint operator on L2(Σ).


If Σ = Rd a natural hypothesis (which if needed can be rephrased in terms of theoriginal metric g) is that :

a(t, x, ∂x) = −∑ij

ajk(t, x)∂xj∂xk +∑aj

(t, x)∂xj + r(t, x),

whereC−1ξ2 ≤ ajk(t, x)ξjξk ≤ Cξ2,

|∂mt ∂αxajk|, |∂mt ∂αxaj], ∂mt ∂αx r bounded locally uniformly in t,an assumption of course related to the uniform pdo calculus on Σ. Let us set for s ∈ Rρsu := (uΣs , i

−1∂tuΣs) and Us the solution of the Cauchy problem on Σs, i.e.

(∂2t + a(t))Us = 0, ρs Us = 1l.

10.3.2. Parametrices for the Cauchy problem. Let R 3 t 7→ b(t) is a map with values inlinear operators on L2(Σ) with Domb(t) = H1(Σ), b(t)− b∗(t) bounded, and R 3 t 7→ b(t)

be norm continuous with values in B(H1(Σ), L2(Σ)). We denote by Texp(i´ tsb(σ)dσ) the

strongly continuous group with generator b(t). A routine computation shows that

(∂2t + a(t))Texp(i

ˆ t

s

b(σ)dσ) = 0

if and only if b(t) solves the following Riccati equation :

(10.2) i∂tb(t)− b2(t) + a(t) = 0.

This equation can be solved modulo C∞(R,Ψ−∞(Σ)) by symbolic calculus, which amountsto solving transport equations.

Theorem 10.1. There exists b(t) ∈ C∞(R,Ψ1(Σ)), unique modulo C∞(R,Ψ−∞(Σ)) suchthat

i) b(t) = ε(t) + C∞(R,Ψ0(Σ)),

ii) (b(t) + b∗(t))−1 = ε(t)−12 (1l + r−1(t))ε(t)−

12 , r−1(t) ∈ C∞(R,Ψ−1(Σ)),

iii) (b(t) + b∗(t))−1 ≥ c(t)ε(t)−1, for some c(t) > 0

iv) i∂tb− b2 + a = r−∞ ∈ C∞(R,Ψ−∞(Σ)).

Note that if b+(t) := b(t) solves (10.2), so does b−()t := −b∗(t). Setting u±(t, s) =

Texpi´ tsb±(σ)dσ, we obtain the following result.

Theorem 10.2. Setr0±s··= ∓(b+(s)− b−(s))−1b∓(s) ∈ Ψ0(Σ),

r1±s··= ±(b+(s)− b−(s))−1 ∈ Ψ−1(Σ),

r±s f ··= r0±f 0 + r1±f 1, f = (f 0, f 1) ∈ H∞(Σ)⊗ C2.

Then

Us = u+(·, s)r+s + u−(·, s)r−s + C∞(R,Ψ−∞(Σ)).

34 C. GÉRARD

10.3.3. Pure Hadamard states. The following proposition summarizes conditions implyingthat a pair of operators λ±s are the Cauchy surface covariances (at time s) of a (pure)Hadamard state.

Proposition 10.3. Let λ±s : H∞(Σ) ⊗ C2 → H∞(Σ) ⊗ C2 be continuous. Then λ±s arethe Cauchy surface covariances of a Hadamard state ω if :

i) λ±∗s = λ±s , λ± ≥ 0,

ii) λ+s − λ−s = q,

iii) f ∈ H−∞(Σ)⊗ C2 ∩Kerλ∓ ⇒ WF(Usf) ⊂ N±.

If additionally c±s := ±iq−1 λ±s are projections, then ω is pure.

Note that if f ∈ H−∞(Σ)⊗C2 and r∓s f = 0, then WFUsf ⊂ N±. This allows easily toconstruct a pure Hadamard state associated to the asymptotic solution b(t) of (10.2).

In fact if we set :

Ts(b) = (b(s) + b∗(s))−12

(b∗(s) 1lb(s) 1l

),

we note that

q =

(0 1l1l 0

)= Ts(b)

∗(

1l 00 1

)Ts(b).

It follows that

λ±(s) := Ts(b)∗π±Ts(b), π

+ =

(1l 00 0

), π− =

(0 00 1

),

are the Cauchy surface covariances of a pure Hadamard state.

Remark 10.4. One can show that if the Cauchy surface covariances of a state ω arepseudodifferential at some time s, the same is true at any other time s′.

Moreover one can show that any pure Hadamard state with pseudodifferential Cauchysurface covariances is of the form above, for some t 7→ b(t) as in Thm. 10.1

The above construction of Hadamard states by pseudodifferential calculus can be gene-ralized to more delicate situations, like linearized Yang-Mills fields, where the deformationargument cannot be applied anymore, see [GW2].

10.4. Construction of Hadamard states by characteristic Cauchy problem. Itis possible to construct Hadamard states by replacing the (space-like) Cauchy surface Σby some null hypersurface C, typically C is chosen to be the forward lightcone from somepoint p ∈ M . Then the interior M0 of C, when equipped with g is a globally hyperbolicspacetime in its own right. This approach was introduced by Valter Moretti [Mo1], andgeneralized afterwards to various similar situations, [DMP1, DMP2, DS, Mo2] in orderto construct a distinguished Hadamard state on asymptotically flat spacetimes, for theconformal wave equation.

In [GW3] we construct a large family of pure Hadamard states on the cone C, usingagain pseudodifferential calculus.


10.4.1. The geometric framework. Let (M, g) a globally hyperbolic spacetime, p ∈ M adistinguished point of M . Let

C ··= ∂J+(p)\pbe the future lightcone from p and

M0 = I+(p) its interior.

One can show that (M0, g) is also a globally hyperbolic spacetime. Moreover one can showthat :

(10.3) J+(K;M0) = J+(K;M), J−(K;M0) = J−(K;M) ∩M0, ∀ K ⊂M0.

Some global conditions are needed to avoid singularities of C. One assumes that thereexists f ∈ C∞(M) such that :

(1) C ⊂ f−1(0), ∇af 6= 0 on C, ∇af(p) = 0, ∇a∇bf(p) = −2gab(p),

(2) the vector field ∇af is complete on C.

Note that since C is a null hypersurface the vector field ∇af is tangent to C.From this hypothesis it is easy to construct coordinates (f, s, θ) near C, with f, s ∈ R,

θ ∈ Sd−1 such that C ⊂ f = 0 and(10.4) gC = −2dfds+ h(s, θ)dθ2,

where h(s, θ)dθ2 is a Riemannian metric on Sd−1.Such choice of coordinates allows one to identify C with C ··= R × Sd−1. A natural

space of smooth functions on C is then provided by H∞(C) — the intersection of Sobolevspaces of all orders, defined using the round metric m(θ)dθ2 on Sd−1.

10.4.2. Bulk to boundary correspondence. Let us consider the restriction P0 of the Klein-Gordon operator P to C∞(M0), and E0± its advanced/retarded Green’s functions. From(10.3), one obtains E0± = E±M0×M0 , hence :

E0 = EM0×M0 .

This implies that any solution φ0 ∈ Solsc(P0) uniquely extends to φ ∈ Solsc(P ). In parti-cular its trace φ0C is well defined.

It is convenient to introduce the coordinates (s, θ) on C, and to set :

ρ : Solsc(P0)→ C∞(R× Sd−1)

φ 7→ β−1(s, θ)φC (s, θ),

forβ(s, θ) ··= |m|

14 (θ)|h|−

14 (s, θ),

where h(s, θ)dθ2 is defined in (10.4) and m(θ)dθ2 is the round metric on Sd−1.We equip H∞(C) with the symplectic form :

(10.5) g1σCg2 ··=ˆR×Sd−1

(∂sg1g2 − g1∂sg2)|m|12 (θ)dsdθ, g1, g2 ∈ H(C).

Introducing the charge q ··= iσC we have :

(10.6) g1qg2 = 2(g1|Dsg2)L2(C), g1, g2 ∈ H(C),

36 C. GÉRARD

where Ds = i−1∂s is selfadjoint on L2(C) on its natural domain. Clearly (H∞(C), σC) isa complex symplectic space.

One can show the following result (the second statement follows from Stokes theorem),which summarizes the bulk to boundary correspondence :

Proposition 10.5. (1) ρ maps Solsc(P0) into H(C) ;(2) ρ : (Solsc(P0), σ)→ (H(C), σC) is a monomorphism, i.e. :

ρφ1σCρφ2 = φ1σφ2, ∀φ1, φ2 ∈ Solsc(P0).

10.4.3. Hadamard states on the cone. From Prop. 10.5, we see that any quasi-free stateωC on CCR(H∞(C), σC) generates a quasi-free state ω0 on CCR(C∞0 (M0)/PC∞0 (M0), E0).The task is now to give conditions on ωC implying that ω0 is Hadamard.

Let us denote by x = (r, s, y) the coordinates (f, s, θ) near C and by ξ = (ρ, ση) thedual coordinates. The complex covariances of ωC are denoted by λ± ∈ D′(C × C). Theassociated covariances Λ± of ω0 are then :

Λ± ··= (ρ E0)∗ λ± (ρ E0).

One can show the following result.

Theorem 10.6. Assume that λ± : H∞(C) → H∞(C) and CWF(λ±)′ = WF(λ±)′C

= ∅Then if :

i) WF(λ±)′ ∩ (Y1, Y2) : ±σ1 < 0 or ± σ2 < 0 = ∅,ii) WF(λ±)′ ∩ (Y1, Y2) : ±σ1 > 0 and ± σ2 > 0 ⊂ ∆.

Λ± satisfy (Had).

A state on CCR(H∞(C), σC) satisfying the assumptions of Thm. 10.6 will be called aHadamard state on the cone.

10.4.4. Hadamard states on the cone and pseudodifferential calculus. Recall that q definedin (10.6) equals 2Ds, whose resolvent (2Ds − z)−1 is not an elliptic pseudodifferentialoperator on C, for the usual calculus. However it belongs to a larger bi-homogeneous classΨp1,p2(C), which can be loosely defined as Ψp1(R)⊗Ψp2(Sd−1).

It is then possible to construct Hadamard states on the cone by mimicking the argu-ments in Subsect. 10.3. Note that no construction of a parametrix for the characteristicCauchy problem on C is necessary, since the Hadamard condition on the cone is ratherexplicit.

10.4.5. Purity inside the cone. A natural issue with this construction is whether a purestate on the cone generates a pure state inside. This is not a priori obvious, since the mapρ is not surjective. Nevertheless it is shown in [GW3] that purity is preserved, using theresults of Hörmander [Hö] on the characteristic Cauchy problem. This is the only placewhere the solvability of the characteristic Cauchy problem is important.


Références

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Département de Mathématiques, Université de Paris XI, 91405 Orsay Cedex FranceE-mail address: [email protected]

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